WEBVTT 00:00:00.000 --> 00:00:02.550 MALE SPEAKER: The following content is provided under a 00:00:02.550 --> 00:00:04.370 Creative Commons License. 00:00:04.370 --> 00:00:07.410 Your support will help MIT OpenCourseWare continue to 00:00:07.410 --> 00:00:11.060 offer high quality educational resources for free. 00:00:11.060 --> 00:00:13.960 To make a donation or view additional materials from 00:00:13.960 --> 00:00:18.365 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:18.365 --> 00:00:19.615 ocw.mit.edu. 00:00:24.741 --> 00:00:26.650 PROFESSOR: So today I want to finish up 00:00:26.650 --> 00:00:27.900 thinking about circuits. 00:00:30.090 --> 00:00:32.460 And the major topic for today is just one -- 00:00:32.460 --> 00:00:34.890 thinking about abstractions that we can use for thinking 00:00:34.890 --> 00:00:36.690 about circuits. 00:00:36.690 --> 00:00:40.220 The attractions are things that capitalize on linearity. 00:00:40.220 --> 00:00:42.560 And they include things like [UNINTELLIGIBLE] and 00:00:42.560 --> 00:00:43.810 superposition. 00:00:45.380 --> 00:00:47.430 OK so what I want to do is finish up 00:00:47.430 --> 00:00:48.760 thinking about circuits. 00:00:48.760 --> 00:00:51.900 And just to get you thinking about circuits, let's think 00:00:51.900 --> 00:00:52.770 about where we are. 00:00:52.770 --> 00:00:56.840 Last time we saw that one of the issues in thinking about 00:00:56.840 --> 00:01:02.380 circuit design, is the fact that every part, in principle, 00:01:02.380 --> 00:01:04.040 interacts with every other part. 00:01:04.040 --> 00:01:09.210 Which makes the design process harder than the design process 00:01:09.210 --> 00:01:11.580 was for things like linear systems, where we thought 00:01:11.580 --> 00:01:15.240 about boxes having inputs and outputs. 00:01:15.240 --> 00:01:17.300 In linear systems, when we thought about boxes having 00:01:17.300 --> 00:01:21.260 inputs and outputs, the output didn't necessarily have any 00:01:21.260 --> 00:01:23.230 affect on the input, unless there was an 00:01:23.230 --> 00:01:26.490 explicit path feedback. 00:01:26.490 --> 00:01:28.830 In circuits, there's feedback always. 00:01:28.830 --> 00:01:31.070 There's no way to avoid it. 00:01:31.070 --> 00:01:33.320 And in fact, that coupling can make thinking about the 00:01:33.320 --> 00:01:35.380 circuits very difficult. 00:01:35.380 --> 00:01:38.530 And we introduce that idea by just thinking about, even if 00:01:38.530 --> 00:01:41.110 you wanted to close a switch, you know a circuit, that's 00:01:41.110 --> 00:01:44.760 logically the same as adding a new part. 00:01:44.760 --> 00:01:47.050 And when you do that, that's going to change the currents 00:01:47.050 --> 00:01:49.320 and voltages everywhere. 00:01:49.320 --> 00:01:51.250 So that's kind of a big thing-- that's kind of the 00:01:51.250 --> 00:01:55.500 thing that's different about circuits from what we've been 00:01:55.500 --> 00:01:56.750 thinking about before. 00:01:59.050 --> 00:02:01.330 Because that complicates design, we'd like some way of 00:02:01.330 --> 00:02:02.110 dealing with it. 00:02:02.110 --> 00:02:04.850 And the way we introduced, and the way that you used in the 00:02:04.850 --> 00:02:07.280 lab, so far, has been to use a buffer. 00:02:07.280 --> 00:02:09.620 We can use an op-amp to make a buffer. 00:02:09.620 --> 00:02:13.480 And a buffer has this nice isolation property. 00:02:13.480 --> 00:02:16.010 In this particular circuit, it has the property that it 00:02:16.010 --> 00:02:18.820 copies this voltage to the output, regardless of what's 00:02:18.820 --> 00:02:20.070 at the output. 00:02:22.170 --> 00:02:24.340 That means we could put a switch here, we could change 00:02:24.340 --> 00:02:26.760 the light bulb, we could do anything we wanted to, and we 00:02:26.760 --> 00:02:30.190 would still know, because of the properties of the op-amp, 00:02:30.190 --> 00:02:32.677 we would still know that this voltage is going to be 8 00:02:32.677 --> 00:02:34.730 Volts, regardless of what we put there. 00:02:34.730 --> 00:02:36.890 So that's very nice, that gives us a modularity, that 00:02:36.890 --> 00:02:39.600 gives us a way to design things. 00:02:39.600 --> 00:02:41.990 It let's us design the left-- 00:02:41.990 --> 00:02:44.580 make a circuit that generates 8 Volts without knowing 00:02:44.580 --> 00:02:48.030 precisely what's going to be the ultimate 00:02:48.030 --> 00:02:49.820 circuit that that drives. 00:02:49.820 --> 00:02:53.910 So that's nice, it allows us to do modularity. 00:02:53.910 --> 00:02:57.650 And in many instances, that's a very good solution. 00:02:57.650 --> 00:02:59.310 In fact in the solution-- in the problems that you are 00:02:59.310 --> 00:03:00.320 working on in lab, where you're 00:03:00.320 --> 00:03:03.280 trying to drive a motor-- 00:03:03.280 --> 00:03:06.710 that's an excellent solution. 00:03:06.710 --> 00:03:09.200 It's not always an excellent solution. 00:03:09.200 --> 00:03:12.590 In some sense, it's very expensive. 00:03:12.590 --> 00:03:15.620 An op-amp is a complicated part. 00:03:15.620 --> 00:03:19.860 If you were to look inside an op-amp, there's some two dozen 00:03:19.860 --> 00:03:22.470 transistors in most op-amps. 00:03:22.470 --> 00:03:25.190 So it's not an inexpensive part, especially when you 00:03:25.190 --> 00:03:27.310 think about this kind of a circuit that only has three 00:03:27.310 --> 00:03:28.770 parts to the left. 00:03:28.770 --> 00:03:33.110 The op-amp is actually a more complex device than 00:03:33.110 --> 00:03:35.370 anything else up. 00:03:35.370 --> 00:03:38.700 So op-amps are wonderful, op-amps allow us to make 00:03:38.700 --> 00:03:41.060 buffers, buffers are wonderful, but they're not 00:03:41.060 --> 00:03:44.540 always the best solution for thinking about modularity. 00:03:44.540 --> 00:03:46.940 And, in fact, there's other ways. 00:03:46.940 --> 00:03:49.020 And so that's what we want to think about today is -- other 00:03:49.020 --> 00:03:55.800 ways for achieving modularity in circuit design. 00:03:55.800 --> 00:03:59.830 And the key to thinking about this, is to think about, well, 00:03:59.830 --> 00:04:02.540 what's the worst thing that could happen? 00:04:02.540 --> 00:04:08.440 If I changed this part arbitrarily, just how bad can 00:04:08.440 --> 00:04:09.690 things get? 00:04:13.290 --> 00:04:15.640 So I'll let you answer that. 00:04:15.640 --> 00:04:19.459 Think about that circuit, and assume that this is a 90 Volt 00:04:19.459 --> 00:04:22.680 power supply, 3 Ohm, 6 Ohm, but this can change-- 00:04:22.680 --> 00:04:23.930 R0 can change. 00:04:26.540 --> 00:04:30.650 I've tabulated some values of R0 and putative, corresponding 00:04:30.650 --> 00:04:32.270 values for V0 and I0. 00:04:35.130 --> 00:04:37.920 Are my putative answers right? 00:04:37.920 --> 00:04:40.150 So take a minute, talk to your neighbor, figure 00:04:40.150 --> 00:04:42.710 out how many of these-- 00:04:42.710 --> 00:04:47.220 let's see-- ten blue numbers are right or wrong. 00:05:40.613 --> 00:05:42.470 It's very quiet, you are allowed to talk to people. 00:07:54.460 --> 00:07:56.160 So how many of the numbers are wrong? 00:07:56.160 --> 00:08:00.230 So raise your hand, indicate by number fingers how many 00:08:00.230 --> 00:08:02.790 mistakes are in the table. 00:08:02.790 --> 00:08:03.820 More votes, more votes-- 00:08:03.820 --> 00:08:05.110 come on, come on. 00:08:05.110 --> 00:08:06.900 if you had talked more, you could blame it on your 00:08:06.900 --> 00:08:09.100 neighbor more easily, so talk quickly. 00:08:19.080 --> 00:08:20.330 OK. 00:08:24.070 --> 00:08:25.585 So does everybody agree with their neighbor? 00:08:29.350 --> 00:08:31.100 OK, i don't see a single right answer. 00:08:34.159 --> 00:08:36.429 So take 30 more seconds and think about it again. 00:08:36.429 --> 00:08:38.590 I don't see any right answers. 00:08:38.590 --> 00:08:41.261 So assume your answer's wrong. 00:08:41.261 --> 00:08:42.690 [CHUCKLES] 00:08:42.690 --> 00:08:43.380 Color blind-- 00:08:43.380 --> 00:08:45.180 OK, so how many of the voltages and 00:08:45.180 --> 00:08:46.430 currents are incorrect? 00:09:14.520 --> 00:09:15.950 OK, how about a re-vote? 00:09:15.950 --> 00:09:18.620 So how many of the blue numbers-- 00:09:18.620 --> 00:09:20.890 the currents and voltages, how many of V0 and I0 -- 00:09:20.890 --> 00:09:22.140 how many of those are incorrect? 00:09:25.240 --> 00:09:26.970 OK, not very many votes-- 00:09:26.970 --> 00:09:29.150 I'd say about 80% correct, now. 00:09:29.150 --> 00:09:31.020 That's definitely an improvement. 00:09:31.020 --> 00:09:32.300 What would have to be true? 00:09:32.300 --> 00:09:34.590 If I wanted to prove that some of these numbers were wrong, 00:09:34.590 --> 00:09:36.340 what could I do? 00:09:36.340 --> 00:09:39.490 Give me a condition that would have to be true if the numbers 00:09:39.490 --> 00:09:40.740 were correct. 00:09:44.616 --> 00:09:48.004 AUDIENCE: V0 has to equal I0? 00:09:48.004 --> 00:09:50.140 PROFESSOR: V0 has to equal I0. 00:09:50.140 --> 00:09:53.460 We know that this resistor better be-- 00:09:53.460 --> 00:09:55.080 better obey Ohm's Law. 00:09:55.080 --> 00:09:56.940 That's what we mean by that symbol. 00:09:56.940 --> 00:10:00.750 So it had better be the case that V0 is I0R0. 00:10:00.750 --> 00:10:02.920 So you you'd want this number to be the 00:10:02.920 --> 00:10:04.600 product of that and that. 00:10:04.600 --> 00:10:07.810 So 0 is the product of 0 and 30, that looks good. 00:10:07.810 --> 00:10:13.750 30 is 2 times 15, 36 is 3 times 12, 48 is 6 times 8. 00:10:13.750 --> 00:10:14.320 60 is-- 00:10:14.320 --> 00:10:17.000 well that's a little marginal. 00:10:17.000 --> 00:10:18.630 how about if I rearrange it a little bit, and say 00:10:18.630 --> 00:10:19.880 what if R is V/I? 00:10:23.310 --> 00:10:26.770 If I is 0, that would make the resistor infinity, so there's 00:10:26.770 --> 00:10:29.180 a way of thinking about that last line as correct. 00:10:29.180 --> 00:10:31.160 That's a little funny. 00:10:31.160 --> 00:10:33.280 Maybe the forward way of thinking about it is, well 00:10:33.280 --> 00:10:36.700 what if I made the resistor infinite? 00:10:36.700 --> 00:10:39.060 if I made the resistor infinite, what would be I? 00:10:42.380 --> 00:10:43.330 0. 00:10:43.330 --> 00:10:44.580 And what would be V? 00:10:48.170 --> 00:10:50.175 What would be the voltage difference if 00:10:50.175 --> 00:10:52.887 this were right, then? 00:10:52.887 --> 00:10:53.873 The voltage divider-- 00:10:53.873 --> 00:10:54.859 AUDIENCE: Oh, yeah. 00:10:54.859 --> 00:10:58.063 PROFESSOR: So if there's no current here, then you can use 00:10:58.063 --> 00:10:59.800 the voltage relationship-- the voltage-divider 00:10:59.800 --> 00:11:01.070 relationship here. 00:11:01.070 --> 00:11:02.240 6 over (3 plus 6) times 90 -- 00:11:02.240 --> 00:11:04.290 which is 60. 00:11:04.290 --> 00:11:06.820 So if you use straightforward reasoning saying, what if the 00:11:06.820 --> 00:11:08.150 resistor was infinitely large? 00:11:08.150 --> 00:11:08.990 What would happen? 00:11:08.990 --> 00:11:11.760 Then you would conclude that the bottom line is OK. 00:11:11.760 --> 00:11:12.630 If you choose-- 00:11:12.630 --> 00:11:18.770 if you show that R equals V over I, are you done? 00:11:18.770 --> 00:11:20.800 What else has to be true in order for the 00:11:20.800 --> 00:11:22.050 numbers to be true? 00:11:28.910 --> 00:11:31.370 It's not very hard. 00:11:31.370 --> 00:11:35.306 AUDIENCE: The voltages drop out of the [UNINTELLIGIBLE]. 00:11:35.306 --> 00:11:37.790 The resistor has to be the same as the-- 00:11:37.790 --> 00:11:39.330 PROFESSOR: So the voltages-- 00:11:39.330 --> 00:11:40.300 that's an instance-- 00:11:40.300 --> 00:11:43.310 what you just said-- is an instance of KVL. 00:11:43.310 --> 00:11:46.360 Basically, the voltage around all of the loops better be 0. 00:11:46.360 --> 00:11:48.550 The sum of the voltages around all of the loops better be 0, 00:11:48.550 --> 00:11:51.290 and the sum of the currents in all of those closed surfaces 00:11:51.290 --> 00:11:52.250 better be 0, right? 00:11:52.250 --> 00:11:55.540 KVL better be satisfied everywhere, and KCL better be 00:11:55.540 --> 00:11:57.240 satisfied everywhere. 00:11:57.240 --> 00:11:59.740 So, in particular, we can ask about this node. 00:11:59.740 --> 00:12:01.660 We could say, do the currents flowing into 00:12:01.660 --> 00:12:05.580 that node sum to 0? 00:12:05.580 --> 00:12:08.560 So if you take a line here-- let's take this line-- 00:12:08.560 --> 00:12:10.710 if V0 happened to be-- 00:12:10.710 --> 00:12:14.070 V0 happens to be the voltage here, right? 00:12:14.070 --> 00:12:19.730 So if V0 were 36, then that puts some voltage on this leg. 00:12:19.730 --> 00:12:21.353 36 would what, 54? 00:12:24.490 --> 00:12:26.500 So then you'd have to say-- 00:12:26.500 --> 00:12:28.420 you'd have to compute, then, the current through here and 00:12:28.420 --> 00:12:30.410 the current through there, and see if those 00:12:30.410 --> 00:12:32.790 currents sum to 0. 00:12:32.790 --> 00:12:36.890 And, in fact, if you do those calculations, you if you can 00:12:36.890 --> 00:12:38.830 convince yourself of those are all true. 00:12:38.830 --> 00:12:40.690 So the answer was 0. 00:12:40.690 --> 00:12:43.640 All of those answers were true. 00:12:43.640 --> 00:12:46.320 The point of doing the exercise was to just remind 00:12:46.320 --> 00:12:50.260 you about how you solve circuits, but also to let us 00:12:50.260 --> 00:12:53.090 look at patterns. 00:12:53.090 --> 00:12:55.650 The interesting thing in this problem is the pattern the 00:12:55.650 --> 00:13:00.140 results between the V's and the I's. 00:13:00.140 --> 00:13:06.290 So if I were to make a plot, in fact, those V's and I's all 00:13:06.290 --> 00:13:09.180 fall on a straight line. 00:13:09.180 --> 00:13:12.400 Well that's pretty interesting. 00:13:12.400 --> 00:13:17.720 So if I plot just this V versus this I-- 00:13:17.720 --> 00:13:19.974 so V equals 0, 30, 36, 0, 30, 36, 48, 60 -- 00:13:23.970 --> 00:13:26.380 and the corresponding I's, 30,15,12,6,0 -- 00:13:26.380 --> 00:13:29.360 30,15,12,6,0. 00:13:29.360 --> 00:13:33.190 Those points all fall on a straight line. 00:13:33.190 --> 00:13:36.820 That suggests that there's some pattern here. 00:13:36.820 --> 00:13:39.630 And if there's a pattern, then there might be a way to 00:13:39.630 --> 00:13:40.730 exploit it. 00:13:40.730 --> 00:13:42.720 So that's what I'm trying to develop-- 00:13:42.720 --> 00:13:46.840 is a way to exploit the pattern that results when 00:13:46.840 --> 00:13:50.470 parts of circuits interact. 00:13:50.470 --> 00:13:52.180 The interesting thing that we-- 00:13:52.180 --> 00:13:55.160 so not only is it true that there's a simple pattern, but 00:13:55.160 --> 00:13:59.370 it turns out that the pattern is completely independent of 00:13:59.370 --> 00:14:02.003 the thing that I put on the right. 00:14:02.003 --> 00:14:06.165 The pattern is a property of the circuit to the left. 00:14:08.960 --> 00:14:12.170 One way to convince yourself of that is to substitute-- 00:14:12.170 --> 00:14:17.120 take that resistor out, and put in a voltage source, and 00:14:17.120 --> 00:14:19.540 redo the problem. 00:14:19.540 --> 00:14:23.180 This time, instead of assuming that there's an Ohm's Law-type 00:14:23.180 --> 00:14:26.120 resistor here, assume there's a constant voltage, and that 00:14:26.120 --> 00:14:30.410 constant voltage is adjusted to 0,30,36,48,60. 00:14:30.410 --> 00:14:32.600 If you re-solve that problem, you get 00:14:32.600 --> 00:14:35.930 exactly these same currents. 00:14:35.930 --> 00:14:41.710 The answer continues to fall on exactly the same line. 00:14:41.710 --> 00:14:43.270 So that's a very interesting pattern. 00:14:43.270 --> 00:14:46.800 The idea is that when the circuit on the left interacts 00:14:46.800 --> 00:14:49.430 with the circuit on the right, regardless of what the circuit 00:14:49.430 --> 00:14:53.490 on the right is, you get a simple relationship between 00:14:53.490 --> 00:14:55.730 the voltage and current that comes out of the 00:14:55.730 --> 00:14:56.980 circuit on the left. 00:14:59.570 --> 00:15:02.510 So that motivates the idea that we can think about the 00:15:02.510 --> 00:15:09.260 left-hand circuit as some kind of a generic part. 00:15:09.260 --> 00:15:12.490 We call that generic part a one-port. 00:15:12.490 --> 00:15:14.750 Think about this circuit on the left-- the thing that's in 00:15:14.750 --> 00:15:17.100 the red box-- 00:15:17.100 --> 00:15:19.120 it's got at its terminals-- 00:15:19.120 --> 00:15:22.490 the terminals are the things that poke through the red box. 00:15:22.490 --> 00:15:24.530 First off it has two terminals, two terminals is 00:15:24.530 --> 00:15:26.760 just like all of our other parts. 00:15:26.760 --> 00:15:29.020 And it's like a resistor, it's like a voltage source, it's 00:15:29.020 --> 00:15:33.800 like a current source, it's a two terminal device. 00:15:33.800 --> 00:15:36.460 And just like a voltage source or a resistor or a current 00:15:36.460 --> 00:15:39.480 source, there's some voltage across those terminals, and 00:15:39.480 --> 00:15:43.470 there's some current that flows in those terminals. 00:15:43.470 --> 00:15:45.170 So there's some current that goes in the plus. 00:15:45.170 --> 00:15:47.210 And that same current comes out the minus. 00:15:47.210 --> 00:15:50.080 That'll be true for this circuit, just the same as it's 00:15:50.080 --> 00:15:53.780 true for a resistor or voltage source or anything else, the 00:15:53.780 --> 00:15:57.580 only difference is, the thing that's inside the box-- 00:15:57.580 --> 00:15:59.100 the thing that's inside the one-port-- 00:15:59.100 --> 00:16:03.510 is more complicated than it was for a simple resistor or 00:16:03.510 --> 00:16:05.080 voltage source or current source. 00:16:05.080 --> 00:16:08.550 So what we can think about, is this whole red box just looks 00:16:08.550 --> 00:16:12.320 like a super part. 00:16:12.320 --> 00:16:16.700 So the interesting thing that happens is, this red box 00:16:16.700 --> 00:16:20.300 behaves like a one-port, like a super part. 00:16:20.300 --> 00:16:25.510 And just like a resistor has a relationship between V and I, 00:16:25.510 --> 00:16:29.610 V equals IR, Or a voltage source has some kind of a 00:16:29.610 --> 00:16:32.520 constraint, V equals V0. 00:16:32.520 --> 00:16:34.250 Or a current source has some kind of a 00:16:34.250 --> 00:16:36.460 constraint, I equals I0. 00:16:36.460 --> 00:16:40.740 This funny part has a relationship between V and I 00:16:40.740 --> 00:16:43.340 that's given by that curve. 00:16:43.340 --> 00:16:47.180 In some sense, it's not very different. 00:16:47.180 --> 00:16:48.510 So we think that-- 00:16:48.510 --> 00:16:52.960 so what we want to do now is figure out the rules that 00:16:52.960 --> 00:16:54.780 govern the currents and voltages that 00:16:54.780 --> 00:16:56.030 flow through one-ports. 00:16:58.270 --> 00:17:03.900 And in particular, how special was this straight line thing? 00:17:03.900 --> 00:17:05.990 I mean if they were always a straight line, that would be 00:17:05.990 --> 00:17:08.960 really easy, right? 00:17:08.960 --> 00:17:09.900 So the question-- 00:17:09.900 --> 00:17:12.650 so the next question I'd like to ask is, just how often are 00:17:12.650 --> 00:17:16.680 we expecting to see straight lines there? 00:17:16.680 --> 00:17:20.260 I already said, the primitive elements that we think about-- 00:17:20.260 --> 00:17:23.390 Ohm's Law, resistors, voltage sources, current sources-- 00:17:23.390 --> 00:17:28.109 they have straight line constraints between the 00:17:28.109 --> 00:17:32.170 voltages and currents that they can generate. 00:17:32.170 --> 00:17:33.080 So think about what I'm doing. 00:17:33.080 --> 00:17:35.630 I'm trying to think about a rule that's going to let me 00:17:35.630 --> 00:17:38.370 describe the currents and voltages into that red box 00:17:38.370 --> 00:17:42.340 from the previous slide, much like I would describe the 00:17:42.340 --> 00:17:46.680 voltages and currents in an Ohm's Law a resistor. 00:17:46.680 --> 00:17:48.910 I can tell you the voltage-current relationship, 00:17:48.910 --> 00:17:53.550 V equals IR for an Ohm's resistor, independent of what 00:17:53.550 --> 00:17:56.190 it's connected to. 00:17:56.190 --> 00:17:57.850 That's the reasoning that I'm using here. 00:17:57.850 --> 00:18:00.450 I'm going to try to figure out, independent of what it is 00:18:00.450 --> 00:18:03.150 connected to, what will be the voltage-current relationship 00:18:03.150 --> 00:18:05.640 for the red box? 00:18:05.640 --> 00:18:10.510 So the question is, when are we expecting straight lines, 00:18:10.510 --> 00:18:13.360 and when are we not expecting straight lines? 00:18:13.360 --> 00:18:15.480 So here's a simple circuit. 00:18:15.480 --> 00:18:18.650 Here's a super part made out of one linear resistor, one 00:18:18.650 --> 00:18:21.090 Ohm's Law resistor, and one voltage source. 00:18:21.090 --> 00:18:24.680 What's the current-voltage relationship for that part? 00:18:24.680 --> 00:18:29.280 What if I put a red box around the whole thing, and asked you 00:18:29.280 --> 00:18:32.920 to draw the I-V curve, the current voltage curve? 00:18:32.920 --> 00:18:35.120 Which would it look like, A,B,C,D -- 00:18:35.120 --> 00:18:36.670 which should have been (1), (2), (3), (4) -- 00:18:36.670 --> 00:18:41.380 so you can raise fingers, or none of the above? 00:18:41.380 --> 00:18:42.440 So take 30 seconds. 00:18:42.440 --> 00:18:44.985 Figure out what would be the I-V curve for this part. 00:21:20.050 --> 00:21:24.850 So if you map A to D into (1) to (4) -- 00:21:24.850 --> 00:21:28.290 which of those plots describes the current-voltage 00:21:28.290 --> 00:21:29.610 relationship for that circuit? 00:21:33.230 --> 00:21:34.890 Map A to D to (1) to (4). 00:21:37.480 --> 00:21:42.890 OK, you're quiet, and the success rate is smaller than 00:21:42.890 --> 00:21:46.240 usual, about 50% correct. 00:21:46.240 --> 00:21:48.800 Take 30 seconds, reconsider your answer, try to get this 00:21:48.800 --> 00:21:50.050 up to 85% or so. 00:22:13.258 --> 00:23:13.150 [CHATTER] 00:23:13.150 --> 00:23:15.920 So which is the best plot? 00:23:15.920 --> 00:23:21.140 Which plot best characterizes the circuit on the top? 00:23:21.140 --> 00:23:22.620 OK, that's much better. 00:23:22.620 --> 00:23:24.000 That's about 75%. 00:23:24.000 --> 00:23:27.490 Certainly not 100%, but better. 00:23:27.490 --> 00:23:30.970 OK so how do you think about this? 00:23:30.970 --> 00:23:33.960 One way to think about it is special cases. 00:23:33.960 --> 00:23:35.900 Can you think about any special cases that are 00:23:35.900 --> 00:23:37.450 particularly easy to check? 00:23:40.170 --> 00:23:41.680 0 -- 00:23:41.680 --> 00:23:46.295 OK, what equals 0? 00:23:46.295 --> 00:23:47.430 AUDIENCE: V or I -- 00:23:47.430 --> 00:23:48.490 PROFESSOR: V or I -- 00:23:48.490 --> 00:23:48.960 yes. 00:23:48.960 --> 00:23:50.340 So what if you were to make -- 00:23:50.340 --> 00:23:54.120 if you were to make I equal to 0? 00:23:54.120 --> 00:23:55.950 First off, if you made I equal to 0, what's 00:23:55.950 --> 00:23:58.982 special in the plots? 00:23:58.982 --> 00:24:03.240 If I equals 0, then you round up x-axis. 00:24:03.240 --> 00:24:07.470 Sorry, if you make I 0, what will V be? 00:24:07.470 --> 00:24:12.775 If you made I be 0, how big will be V? 00:24:12.775 --> 00:24:14.980 5 Volts, right? 00:24:14.980 --> 00:24:17.740 If you make I be 0, then there's no 00:24:17.740 --> 00:24:19.700 voltage across the resistor. 00:24:19.700 --> 00:24:22.280 So the total voltage here will be the same as the voltage 00:24:22.280 --> 00:24:25.690 across the voltage source, V equals 5. 00:24:25.690 --> 00:24:29.310 So the intersection on the x-axis should be at the point 00:24:29.310 --> 00:24:30.610 V equals 5. 00:24:30.610 --> 00:24:32.780 Makes sense? 00:24:32.780 --> 00:24:36.400 Now if you said V equals 0, how big would I be? 00:24:40.810 --> 00:24:46.660 If you set V equal to 0 -- how do you set V equal to 0? 00:24:46.660 --> 00:24:47.654 AUDIENCE: Negative 2.5. 00:24:47.654 --> 00:24:51.133 PROFESSOR: Negative 2.5, so what is negative 2.5? 00:24:51.133 --> 00:24:53.630 AUDIENCE: The value of I. 00:24:53.630 --> 00:24:55.416 PROFESSOR: So do you need-- 00:24:55.416 --> 00:24:59.900 how do we set V equal to 0? 00:24:59.900 --> 00:25:04.682 What's the circuit way of saying, set V equal to 0? 00:25:09.030 --> 00:25:11.530 You set V equals 0 in a circuit by-- 00:25:11.530 --> 00:25:12.450 AUDIENCE: Grounding. 00:25:12.450 --> 00:25:14.030 PROFESSOR: --grounding, by setting it-- 00:25:14.030 --> 00:25:15.540 by putting in a wire. 00:25:15.540 --> 00:25:18.516 So you run a wire from here to here. 00:25:18.516 --> 00:25:20.430 Voltage across a wire is always 0, right? 00:25:20.430 --> 00:25:22.680 So you set something to 0 by putting a wire across it. 00:25:22.680 --> 00:25:25.220 We call that a short circuit. 00:25:25.220 --> 00:25:27.990 OK, there's a short path for the voltage travel. 00:25:27.990 --> 00:25:34.100 So you put a short circuit here, and then I becomes V 00:25:34.100 --> 00:25:40.330 over R, and the only trick is that it's up, right? 00:25:40.330 --> 00:25:43.050 Current likes to flow down the electrochemical gradient. 00:25:43.050 --> 00:25:46.760 So it likes to go down the electrical gradient. 00:25:46.760 --> 00:25:50.580 So the current is going to go up, but the reference 00:25:50.580 --> 00:25:56.790 direction for this I is down, so I is minus 2.5. 00:25:56.790 --> 00:26:01.310 So the intersection on this axis is minus 2.5. 00:26:01.310 --> 00:26:03.710 So we know that it has to go through a negative on the 00:26:03.710 --> 00:26:09.130 bottom, and it has to go through a positive V on the 00:26:09.130 --> 00:26:12.670 x-axis, and the only curve that does that is (A). 00:26:12.670 --> 00:26:15.180 And if we wanted to be a little bit more fancy, we 00:26:15.180 --> 00:26:18.210 could figure out the general rule. 00:26:18.210 --> 00:26:20.870 We could just write an expression for Vr. 00:26:20.870 --> 00:26:24.040 Well by KVL, Vr is always the difference 00:26:24.040 --> 00:26:27.190 between V and 5 Volts. 00:26:27.190 --> 00:26:29.040 And we could write an expression for the current 00:26:29.040 --> 00:26:29.780 through the resistor. 00:26:29.780 --> 00:26:34.730 That's just V over R. Vr is V minus 5, R is 2. 00:26:34.730 --> 00:26:37.770 So I get some expression which is, lo and behold, linear in 00:26:37.770 --> 00:26:42.660 V. So just like that more complicated circuit that I'd 00:26:42.660 --> 00:26:46.860 looked at, I ended up with a straight line relationship in 00:26:46.860 --> 00:26:47.990 the current voltage-- 00:26:47.990 --> 00:26:49.980 the relationship between the current and voltage falls on a 00:26:49.980 --> 00:26:52.990 straight line. 00:26:52.990 --> 00:26:55.160 So how special is that? 00:26:55.160 --> 00:26:58.770 Well it actually happens pretty robustly. 00:26:58.770 --> 00:27:01.990 Think about what would happen if I had two parts, the 00:27:01.990 --> 00:27:03.130 generically-- 00:27:03.130 --> 00:27:06.840 so I'm thinking about these being generic boxes, inside 00:27:06.840 --> 00:27:07.710 could be anything. 00:27:07.710 --> 00:27:09.430 There could be a current source, a voltage source, a 00:27:09.430 --> 00:27:13.060 linear resistor, or some other one-port. 00:27:13.060 --> 00:27:15.590 So if I had a generic box here, and a generic box here, 00:27:15.590 --> 00:27:20.320 and I connected them in parallel, under the condition 00:27:20.320 --> 00:27:25.020 that each of the generic boxes had a straight line 00:27:25.020 --> 00:27:29.060 current-voltage relationship, it's easy to argue that the 00:27:29.060 --> 00:27:32.210 resulting relationship between current and voltage for the 00:27:32.210 --> 00:27:35.940 parallel combination is also a straight line. 00:27:35.940 --> 00:27:39.060 All you need to do is realize that if you hook two things in 00:27:39.060 --> 00:27:44.300 parallel, the parallel voltage is the same as V1 and V2. 00:27:44.300 --> 00:27:46.210 So all the voltages are the same. 00:27:46.210 --> 00:27:47.220 And the currents add. 00:27:47.220 --> 00:27:50.270 So Ip, the current end of the parallel combination is the 00:27:50.270 --> 00:27:51.520 sum of the two I's. 00:27:53.760 --> 00:27:56.860 So if you think about the relationship between V1 and 00:27:56.860 --> 00:28:00.730 I1, and the relationship between V2 and I2, you could 00:28:00.730 --> 00:28:02.760 derive the relationship between Ip 00:28:02.760 --> 00:28:07.020 and Vp by just adding. 00:28:07.020 --> 00:28:11.710 V1 equals v2 equals Vp, that's what we saw here. 00:28:11.710 --> 00:28:13.650 And the current at the bottom is the sum of 00:28:13.650 --> 00:28:14.900 the other two currents. 00:28:17.480 --> 00:28:20.170 So, in particular, if this was a straight line, and that was 00:28:20.170 --> 00:28:22.490 a straight line, the sum of two straight lines is a 00:28:22.490 --> 00:28:24.830 straight line, right? 00:28:24.830 --> 00:28:28.090 If you add two straight lines, you get a new straight line. 00:28:28.090 --> 00:28:31.200 So what this shows is that if you started with parts that 00:28:31.200 --> 00:28:34.630 were themselves straight lines, parallel combinations 00:28:34.630 --> 00:28:39.960 would generate a new part with another straight line. 00:28:39.960 --> 00:28:42.390 Same sort of thing happens if the two parts we're in series. 00:28:45.200 --> 00:28:48.100 In series we would have Iseries -- 00:28:48.100 --> 00:28:50.820 it's the same as I1 equals I2-- 00:28:50.820 --> 00:28:54.300 so that's the equivalent of the y-axes, where previously 00:28:54.300 --> 00:28:55.660 we had equivalents of the x-axes. 00:28:58.370 --> 00:29:03.250 And now if both of these boxes had linear I-V curves, I would 00:29:03.250 --> 00:29:07.290 now add horizontally rather than adding vertically. 00:29:07.290 --> 00:29:08.840 But I have the same result. 00:29:08.840 --> 00:29:13.470 If the two, individual one-ports had linear I-V 00:29:13.470 --> 00:29:19.280 curves, and if I add horizontally, I'll get a new 00:29:19.280 --> 00:29:20.750 linear I-V curve. 00:29:23.642 --> 00:29:26.910 And in fact, if you start-- 00:29:26.910 --> 00:29:31.830 if you put any combination of parts that have linear I-V 00:29:31.830 --> 00:29:37.350 curves together to form a new circuit-- 00:29:37.350 --> 00:29:41.470 a new one-port, the I-V curve for that new one-port will be 00:29:41.470 --> 00:29:43.280 a linear function. 00:29:43.280 --> 00:29:46.530 And the way to see that, is to think about linear equations. 00:29:49.160 --> 00:29:52.570 Remember when we solve a circuit, we have to have one 00:29:52.570 --> 00:29:57.385 law for every element, Like Ohm's Law-- it's V equals IR, 00:29:57.385 --> 00:30:00.480 or voltage source is V equals V0 or something like that-- 00:30:00.480 --> 00:30:03.970 we need one law for every element. 00:30:03.970 --> 00:30:07.570 And then we have KVL and KCL. 00:30:07.570 --> 00:30:11.310 Well all of the-- if we start with the assumption that each 00:30:11.310 --> 00:30:14.210 component has a straight line relationship between voltage 00:30:14.210 --> 00:30:16.760 and current, which is true for linear resistors, voltage 00:30:16.760 --> 00:30:19.000 sources, and current sources. 00:30:19.000 --> 00:30:23.020 Those equations that describe those parts are what we call 00:30:23.020 --> 00:30:25.270 linear equations. 00:30:25.270 --> 00:30:33.570 Linear equation is an equation where the function of the 00:30:33.570 --> 00:30:35.760 unknowns is a quote -- 00:30:35.760 --> 00:30:36.950 linear function. 00:30:36.950 --> 00:30:38.920 It has the form-- 00:30:38.920 --> 00:30:42.750 some constant times an unknown, plus some other 00:30:42.750 --> 00:30:46.140 constant times some other unknown, plus, in principle, 00:30:46.140 --> 00:30:48.310 any number of those. 00:30:48.310 --> 00:30:49.923 But it has to be a linear function, so you're not 00:30:49.923 --> 00:30:52.950 allowed to have things like V-squared in there. 00:30:52.950 --> 00:30:55.480 But if you think about things like voltage sources and 00:30:55.480 --> 00:30:59.130 current sources and Ohm's Law, they don't have squares in 00:30:59.130 --> 00:31:02.180 there, they're linear equations. 00:31:02.180 --> 00:31:05.000 And the idea then, when we're solving for the I-V 00:31:05.000 --> 00:31:08.810 relationship for this new one-port thing, all we're 00:31:08.810 --> 00:31:12.760 doing is we're solving a system of linear equation. 00:31:12.760 --> 00:31:16.820 Sort of in the abstract, the idea is that we write down all 00:31:16.820 --> 00:31:19.140 of the component equations inside the box, we write down 00:31:19.140 --> 00:31:22.570 all of the relevant KVL and KCL, and then we solve. 00:31:22.570 --> 00:31:24.570 So they're all going to have-- 00:31:24.570 --> 00:31:26.850 each one of those equations is going to be linear, so it's 00:31:26.850 --> 00:31:32.200 going to be something like, say, a0x0 if x is my unknown. 00:31:32.200 --> 00:31:35.270 'x' could be a current or a voltage. 00:31:35.270 --> 00:31:40.080 And then I might have sum of x1, then I might have sum of 00:31:40.080 --> 00:31:43.020 x2, and I might have a whole bunch of things, and they all 00:31:43.020 --> 00:31:48.790 add together to be some constant, like that. 00:31:48.790 --> 00:31:51.550 So that might represent part one-- 00:31:51.550 --> 00:31:55.270 that might represent one of the components in the box. 00:31:55.270 --> 00:31:57.310 Then I would have another component, which would be some 00:31:57.310 --> 00:31:59.075 other linear equation. 00:32:06.490 --> 00:32:10.930 Then I'd have a KCL equation, well KCL is easy, because that 00:32:10.930 --> 00:32:13.410 only has currents in it, and the multipliers are 00:32:13.410 --> 00:32:15.260 all 1 or minus 1. 00:32:15.260 --> 00:32:16.670 Right, so that's clearly linear. 00:32:16.670 --> 00:32:18.300 And I have KVL equations. 00:32:18.300 --> 00:32:19.840 Those are linear. 00:32:19.840 --> 00:32:24.260 And the point is, that when you solve linear equations, 00:32:24.260 --> 00:32:26.240 you get a new linear equation. 00:32:26.240 --> 00:32:30.200 So think about solving this by the method of substitution. 00:32:30.200 --> 00:32:34.060 I could figure out what is x0 here. 00:32:34.060 --> 00:32:36.860 If I use this equation to figure out x0. 00:32:36.860 --> 00:32:41.000 x0 would be a linear function of all the other x's. 00:32:41.000 --> 00:32:45.290 So then if I plug that linear function into here, I replace 00:32:45.290 --> 00:32:48.420 this linear equation with a new linear equation with one 00:32:48.420 --> 00:32:50.530 fewer unknown. 00:32:50.530 --> 00:32:53.980 if I keep doing that, I just keep replacing linear 00:32:53.980 --> 00:32:55.780 equations with other linear equations. 00:32:55.780 --> 00:32:59.350 When I'm all done, I'm left with a new linear equation. 00:32:59.350 --> 00:33:05.030 That's why, if you start with parts that linear I-V curves, 00:33:05.030 --> 00:33:08.230 you'll end up with a straight line I-V curve. 00:33:08.230 --> 00:33:12.520 So that idea that the I-V function is a straight line is 00:33:12.520 --> 00:33:13.380 quite robust. 00:33:13.380 --> 00:33:18.420 It will happen anytime you go to circuit out of ideal 00:33:18.420 --> 00:33:21.360 parts-- whereby ideal parts, I mean Ohm's Law resistors, 00:33:21.360 --> 00:33:25.190 voltage sources, and current sources. 00:33:25.190 --> 00:33:30.310 So that has a very interesting circuit interpretation. 00:33:30.310 --> 00:33:34.160 If I know that an arbitrary circuit can be represented by 00:33:34.160 --> 00:33:35.090 a straight line-- 00:33:35.090 --> 00:33:36.680 if the I-V curve can be represented 00:33:36.680 --> 00:33:38.350 by a straight line-- 00:33:38.350 --> 00:33:43.180 well that generates an equivalent set. 00:33:43.180 --> 00:33:46.160 There's obviously more than one circuit that could 00:33:46.160 --> 00:33:48.960 generate the same straight line. 00:33:48.960 --> 00:33:53.480 Here's a circuit that can generate that straight line. 00:33:53.480 --> 00:33:56.610 In fact, it will always be true that I can generate a 00:33:56.610 --> 00:33:59.870 circuit with one voltage source and one Ohm's Law 00:33:59.870 --> 00:34:05.130 resistor that will mimic the behavior of any arbitrary 00:34:05.130 --> 00:34:07.630 combination of resistors, voltage 00:34:07.630 --> 00:34:09.940 sources, and current sources. 00:34:09.940 --> 00:34:11.850 All I need to do is think about, you take the 00:34:11.850 --> 00:34:17.170 complicated thing, figure out its straight line plot, now 00:34:17.170 --> 00:34:20.909 you read off some critical numbers from this plot. 00:34:20.909 --> 00:34:25.500 You say, OK, well what if the current were 0? 00:34:25.500 --> 00:34:28.630 Well if the current were 0, I'd be on this axis. 00:34:28.630 --> 00:34:34.320 If the current were 0 in the circuit, V would be V0. 00:34:34.320 --> 00:34:37.110 So that means you look over here, you figure out the 00:34:37.110 --> 00:34:41.020 x-intercept, and the x-intercept is the value of 00:34:41.020 --> 00:34:42.270 the voltage source. 00:34:46.130 --> 00:34:52.190 Similarly, if you figure out the rate of growth of I -- 00:34:52.190 --> 00:34:57.240 so more generally, if you solve for I, I will be the 00:34:57.240 --> 00:35:01.110 difference between V and V0 divided by R, that's just 00:35:01.110 --> 00:35:04.540 Ohm's law for the resistor. 00:35:04.540 --> 00:35:09.580 And that then lets you figure out the law for the slope. 00:35:09.580 --> 00:35:12.710 The slope over here is going to turn out to be one over 00:35:12.710 --> 00:35:13.260 this resistor-- 00:35:13.260 --> 00:35:15.130 I should have had-- this should be R0, 00:35:15.130 --> 00:35:16.530 these two should match. 00:35:16.530 --> 00:35:22.170 So the slope of this line is one over that resistor. 00:35:22.170 --> 00:35:25.980 So in general, regardless of how complicated the box is, 00:35:25.980 --> 00:35:29.110 figure out the straight line and I-V curves, read off the 00:35:29.110 --> 00:35:33.070 x-axis, read off the slope, and that lets you construct a 00:35:33.070 --> 00:35:36.590 simple circuit that has the same Vi curve. 00:35:36.590 --> 00:35:40.390 We call that circuit the Thevenin Equivalent. 00:35:40.390 --> 00:35:44.470 What that means is, you can think about a complicated 00:35:44.470 --> 00:35:47.980 circuit, regardless of how many parts, by a circuit it 00:35:47.980 --> 00:35:50.270 just has two. 00:35:50.270 --> 00:35:54.620 That's an abstraction that lets us think more simply 00:35:54.620 --> 00:36:00.230 about complicated circuits even if there's no buffers. 00:36:00.230 --> 00:36:03.990 This is true always. 00:36:03.990 --> 00:36:05.650 OK, it's not true-- 00:36:05.650 --> 00:36:08.980 it's not it's not the case that if I change the thing 00:36:08.980 --> 00:36:10.460 that I put here-- 00:36:10.460 --> 00:36:13.550 if I change the thing I put here, this relationship is 00:36:13.550 --> 00:36:14.350 still true. 00:36:14.350 --> 00:36:20.900 It's the equivalent of Ohm's Law for complicated circuits. 00:36:20.900 --> 00:36:23.010 Ohm's Law is what I get if what's in the 00:36:23.010 --> 00:36:26.050 box is a single resistor. 00:36:26.050 --> 00:36:28.760 If what's in the box is complicated, more generally 00:36:28.760 --> 00:36:33.170 this is what I get, the Thevenin Equivalent. 00:36:33.170 --> 00:36:34.970 Of course there's lots of circuits that 00:36:34.970 --> 00:36:38.380 have this I-V curve. 00:36:38.380 --> 00:36:41.320 So a different one is a current 00:36:41.320 --> 00:36:43.490 source with a resistor. 00:36:43.490 --> 00:36:46.570 That's called a Norton Equivalent. 00:36:46.570 --> 00:36:47.930 You do the same sort of thing. 00:36:47.930 --> 00:36:52.190 What would happen if here, if I set V to be 0? 00:36:52.190 --> 00:36:54.030 Well how do I make V be 0? 00:36:54.030 --> 00:36:55.980 I make V be 0 -- 00:36:55.980 --> 00:36:57.720 Did I write that on the slide? 00:36:57.720 --> 00:36:58.520 no I didn't write it on the slide. 00:36:58.520 --> 00:37:01.480 I make V be 0 by putting a short circuit here. 00:37:01.480 --> 00:37:03.220 I connect a wire. 00:37:03.220 --> 00:37:07.450 If I put a wire here, then how big is the current I? 00:37:10.710 --> 00:37:14.870 Well if I put a wire here, it's easier for this current 00:37:14.870 --> 00:37:17.300 to go through the wire than it is to go 00:37:17.300 --> 00:37:19.600 through that resistor. 00:37:19.600 --> 00:37:23.580 So all of this current goes through the wire. 00:37:23.580 --> 00:37:26.800 All of this current is I0. 00:37:26.800 --> 00:37:30.570 So if I put a short circuit across V -- 00:37:30.570 --> 00:37:34.660 if I make V be 0, how big is i? 00:37:34.660 --> 00:37:36.430 Minus I0 right? 00:37:36.430 --> 00:37:38.260 All the current I0 goes through, it 00:37:38.260 --> 00:37:41.140 just goes to backwards. 00:37:41.140 --> 00:37:44.400 So that's how I get this point. 00:37:44.400 --> 00:37:49.130 So if I wanted to replace some complicated circuit with a 00:37:49.130 --> 00:37:54.830 Norton Equivalent, I would take the complicated system, 00:37:54.830 --> 00:38:01.270 to figure out the I-V curve, read off the intercept on this 00:38:01.270 --> 00:38:07.030 axis, change the sign-- 00:38:07.030 --> 00:38:09.370 that's the most confusing part, by the way. 00:38:09.370 --> 00:38:12.840 This is the error that you all make. 00:38:12.840 --> 00:38:15.460 There's a minus sign in the current relationship. 00:38:15.460 --> 00:38:19.370 We like to draw-- it just makes us feel good to have the 00:38:19.370 --> 00:38:22.260 arrow go up. 00:38:22.260 --> 00:38:24.420 And if you make the arrow go up, it's in the wrong 00:38:24.420 --> 00:38:30.710 direction to I. So when we did the Thevenin equivalent, there 00:38:30.710 --> 00:38:34.150 was no sign from it, this was V0. 00:38:34.150 --> 00:38:36.700 So the voltage on the voltage source is equal to the 00:38:36.700 --> 00:38:38.140 x-intercept. 00:38:38.140 --> 00:38:41.015 But when we do the Norton, the current in the current source 00:38:41.015 --> 00:38:46.280 is minus the y-intercept. 00:38:46.280 --> 00:38:47.650 Same idea, though. 00:38:47.650 --> 00:38:48.250 OK? 00:38:48.250 --> 00:38:51.590 So the idea is that Thevenin and Norton equivalent circuits 00:38:51.590 --> 00:38:54.150 are equivalent, in the sense that they 00:38:54.150 --> 00:38:57.200 generate the same voltage. 00:38:57.200 --> 00:38:59.160 that the more complicated circuit did. 00:38:59.160 --> 00:39:02.060 So that means for thinking about the circuit we can 00:39:02.060 --> 00:39:06.580 ignore the complicated stuff, and just know two numbers, V0 00:39:06.580 --> 00:39:08.486 and R0, or I0 and R0. 00:39:14.550 --> 00:39:21.060 So one more step, what this all means is, that if you can 00:39:21.060 --> 00:39:25.180 represent the current voltage relationship for an arbitrary 00:39:25.180 --> 00:39:30.120 circuit in terms of a straight line, that means that when 00:39:30.120 --> 00:39:32.820 we're trying to characterize an arbitrary circuit, it 00:39:32.820 --> 00:39:34.580 doesn't matter how complicated it is. 00:39:34.580 --> 00:39:38.480 It could have 100 parts in it. 00:39:38.480 --> 00:39:41.830 Regardless of how complicated it is, I only need to measure 00:39:41.830 --> 00:39:46.060 two things in order to fully characterize it. 00:39:46.060 --> 00:39:49.870 If a circuit is made out of linear resistors, voltage 00:39:49.870 --> 00:39:54.600 sources, and current sources, three special linear parts-- 00:39:54.600 --> 00:39:58.110 if a circuit is composed entirely out of linear parts, 00:39:58.110 --> 00:40:00.490 doesn't matter how many parts are in it, there could 100, 00:40:00.490 --> 00:40:02.130 there could be 1,000-- 00:40:02.130 --> 00:40:05.180 I only need to measure two things to get a complete 00:40:05.180 --> 00:40:06.610 description. 00:40:06.610 --> 00:40:09.800 And that's because two points determine a straight line. 00:40:09.800 --> 00:40:14.820 I know, by having proved it by using linear algebra, I know 00:40:14.820 --> 00:40:17.330 by using linear algebra that the solution 00:40:17.330 --> 00:40:18.530 is a straight line. 00:40:18.530 --> 00:40:20.840 I know from geometry that two points 00:40:20.840 --> 00:40:22.160 determine a straight line. 00:40:22.160 --> 00:40:24.830 I only need to find out two points. 00:40:24.830 --> 00:40:28.250 So by convention, the easiest two points is usually the 00:40:28.250 --> 00:40:29.740 simplest cases. 00:40:29.740 --> 00:40:33.530 Set the voltage to 0 and set the current to be 0. 00:40:33.530 --> 00:40:36.810 So that motivates the idea that if I have an arbitrary 00:40:36.810 --> 00:40:41.240 circuit, if I want to figure out this reduced complexity 00:40:41.240 --> 00:40:42.900 abstraction-- 00:40:42.900 --> 00:40:45.100 say I want to make a Thevenin Equivalent-- 00:40:45.100 --> 00:40:48.460 what I would do is first to ask the question, how big is 00:40:48.460 --> 00:40:52.330 the open circuit voltage? 00:40:52.330 --> 00:40:54.210 Open circuit means there's no 00:40:54.210 --> 00:40:56.420 connection between the terminals. 00:40:56.420 --> 00:40:59.380 That means the current is 0. 00:40:59.380 --> 00:41:01.476 If the current is 0, I'm on the x-axis. 00:41:04.060 --> 00:41:06.610 So over here I'm on the x-axis, so I'm thinking about 00:41:06.610 --> 00:41:09.000 the red point. 00:41:09.000 --> 00:41:12.830 Open circuit over here means there's no connection here, 00:41:12.830 --> 00:41:16.590 which means the current is 0, and all I need to ask is-- 00:41:16.590 --> 00:41:18.960 regardless of how complicated it is the circuit-- 00:41:18.960 --> 00:41:23.100 how big is the voltage that I would measure here? 00:41:23.100 --> 00:41:25.690 So in this circuit, if I have a-- 00:41:29.630 --> 00:41:32.430 yeah, I confused myself for a moment. 00:41:32.430 --> 00:41:37.440 if this current is 0, then this voltage drop is 0. 00:41:37.440 --> 00:41:41.040 This voltage source prescribes the voltage between these 00:41:41.040 --> 00:41:45.930 nodes to be 1 Volt, so V0 is 1 Volt. 00:41:45.930 --> 00:41:47.710 So I just fell one point. 00:41:47.710 --> 00:41:53.480 I set I to be 0, I found the open circuit voltage. 00:41:53.480 --> 00:41:55.440 Then I need to find one other point, because I 00:41:55.440 --> 00:41:56.740 only to find two. 00:41:56.740 --> 00:42:00.590 So I'll find the short circuit current. 00:42:00.590 --> 00:42:03.540 Imagine that I put a wire between the input and the 00:42:03.540 --> 00:42:07.230 output, and I'll compute how much current that circuit 00:42:07.230 --> 00:42:09.870 generates in that wire. 00:42:09.870 --> 00:42:12.640 Again the only confusing part is that the reference 00:42:12.640 --> 00:42:14.540 directions are backwards to the way you might have 00:42:14.540 --> 00:42:16.220 expected them to be. 00:42:16.220 --> 00:42:20.020 If I put a short circuit here, then the voltage between these 00:42:20.020 --> 00:42:21.400 nodes is still 1 Volt. 00:42:21.400 --> 00:42:24.540 That's what the voltage source always says. 00:42:24.540 --> 00:42:29.100 So the current that flows through this wire, the short 00:42:29.100 --> 00:42:34.160 circuit current, is just this V over that R, except it's in 00:42:34.160 --> 00:42:35.410 the negative direction. 00:42:38.640 --> 00:42:41.320 And the way you can think about that is that the slope 00:42:41.320 --> 00:42:43.460 of this curve has to be positive. 00:42:43.460 --> 00:42:45.710 I need-- because of the way Ohm's Law works-- 00:42:45.710 --> 00:42:48.780 increasing the voltage better tends to increase the current, 00:42:48.780 --> 00:42:51.090 because that's what Ohm's Law resistors do. 00:42:51.090 --> 00:42:54.230 So I need to have this slope be positive if it's going to 00:42:54.230 --> 00:42:56.720 be Ohm's Law. 00:42:56.720 --> 00:43:00.350 So I characterize just those two points, and then the 00:43:00.350 --> 00:43:04.570 resistance is simply the ratio of the two. 00:43:04.570 --> 00:43:07.560 So the resistance is related to the slope. 00:43:07.560 --> 00:43:09.240 It's 1 over the slope, you don't need 00:43:09.240 --> 00:43:10.380 to worry about that. 00:43:10.380 --> 00:43:13.250 The resistance is always V over I, so you just take the 00:43:13.250 --> 00:43:15.900 open circuit voltage and divide by the short circuit 00:43:15.900 --> 00:43:20.150 current, minus sign, and you get that the 00:43:20.150 --> 00:43:21.530 resistor must be 2 Ohms. 00:43:21.530 --> 00:43:23.200 Is that all clear? 00:43:23.200 --> 00:43:26.760 The idea is that we're trying to build an abstraction that 00:43:26.760 --> 00:43:31.000 lets us simplify the way we think about circuits, without 00:43:31.000 --> 00:43:32.250 introducing buffers. 00:43:34.450 --> 00:43:39.900 So that means then that these two circuits are equivalent to 00:43:39.900 --> 00:43:43.110 that circuit, in the sense that they all share 00:43:43.110 --> 00:43:45.110 the same I-V curve. 00:43:45.110 --> 00:43:47.640 If you substituted one server for the other, you couldn't 00:43:47.640 --> 00:43:52.230 tell from outside the red box which was on the 00:43:52.230 --> 00:43:56.970 inside of the red box. 00:43:56.970 --> 00:43:58.830 OK so I'll do an example now. 00:43:58.830 --> 00:44:01.050 Think about, what if I want to find the equivalent-- 00:44:01.050 --> 00:44:03.620 the Thevenin equivalent for this circuit. 00:44:03.620 --> 00:44:06.200 I just do the things I just told you to do. 00:44:06.200 --> 00:44:09.910 First thing I think about is, what's the 00:44:09.910 --> 00:44:12.535 open circuit voltage? 00:44:12.535 --> 00:44:14.740 So If I think about open circuit, there's 00:44:14.740 --> 00:44:17.290 no connection here. 00:44:17.290 --> 00:44:20.430 That means this current is 0. 00:44:20.430 --> 00:44:22.450 That means that the voltage that develops 00:44:22.450 --> 00:44:25.140 is the voltage divider. 00:44:25.140 --> 00:44:26.670 So I get 7 and a 1/2 volts-- 00:44:26.670 --> 00:44:31.750 3 over (1 plus 3) times 10, right. 00:44:31.750 --> 00:44:33.660 So that gives me one point-- 00:44:33.660 --> 00:44:36.620 that tells me the voltage source for the Thevenin 00:44:36.620 --> 00:44:37.940 equivalent. 00:44:37.940 --> 00:44:40.580 Then for the second point, I want to think about the short 00:44:40.580 --> 00:44:41.790 circuit current. 00:44:41.790 --> 00:44:46.550 So I consider putting a wire here, and then I compute the 00:44:46.550 --> 00:44:49.990 amount of current that flows in that wire. 00:44:49.990 --> 00:44:54.610 And in this circuit, this wire shorts out that resistor, so 00:44:54.610 --> 00:44:57.180 all the current goes through this wire, and none of the 00:44:57.180 --> 00:44:59.720 current goes through that resistor. 00:44:59.720 --> 00:45:02.450 So that means that the total current that flows is 10 Volts 00:45:02.450 --> 00:45:06.870 divided by 1 Ohm, which is 10 Amps. 00:45:06.870 --> 00:45:09.190 And then I know that the equivalent resistance is the 00:45:09.190 --> 00:45:11.980 ratio of the open circuit voltage to the short circuit 00:45:11.980 --> 00:45:15.630 current, except I have to worry about the minus sign. 00:45:15.630 --> 00:45:19.190 And so I end up with the equivalent resistance being 00:45:19.190 --> 00:45:22.100 the 7.5 Volts, which is the open circuit voltage, divided 00:45:22.100 --> 00:45:25.760 by 10 Amps, so I get 7.5 Ohms. 00:45:25.760 --> 00:45:29.280 And so the answer then, is that here is the circuit I 00:45:29.280 --> 00:45:31.970 started with, here's the Thevenin equivalent circuit, 00:45:31.970 --> 00:45:33.970 they're identical in the sense that they have 00:45:33.970 --> 00:45:36.390 the same I-V curve. 00:45:36.390 --> 00:45:39.300 And you can just sort of see why that has to be true. 00:45:39.300 --> 00:45:43.020 If you think about, here's the two circuits, and if you think 00:45:43.020 --> 00:45:47.490 about the simple cases, if you set I to be 0, the voltages 00:45:47.490 --> 00:45:49.590 better be the same. 00:45:49.590 --> 00:45:52.800 Well over here it's 7.5 by the voltage divider, over here 00:45:52.800 --> 00:45:55.155 it's 7.5 by the fact that there's no current going 00:45:55.155 --> 00:45:57.460 through that resistor. 00:45:57.460 --> 00:46:00.050 And the short circuit current better be the same. 00:46:00.050 --> 00:46:02.080 So over here, if you short this out, you're going to get 00:46:02.080 --> 00:46:04.090 10 Amps, over here if you short it out, you're going to 00:46:04.090 --> 00:46:05.160 get 10 Amps. 00:46:05.160 --> 00:46:06.550 So you can always go back and forth. 00:46:06.550 --> 00:46:09.680 The point is, that you can substitute the simpler circuit 00:46:09.680 --> 00:46:12.350 for the more complex circuit, because they have 00:46:12.350 --> 00:46:15.160 the same I-V curve. 00:46:15.160 --> 00:46:21.120 OK, just to make sure that you're following me, here's a 00:46:21.120 --> 00:46:26.780 question that has to do with taking the same circuit, but 00:46:26.780 --> 00:46:29.950 considering the Thevenin equivalent-- 00:46:29.950 --> 00:46:31.980 Thevenin or Norton equivalent-- 00:46:31.980 --> 00:46:33.350 at three different ports. 00:46:33.350 --> 00:46:35.300 Here I'm thinking about what would happen if I looked in 00:46:35.300 --> 00:46:41.380 terminal A, what if I looked in terminal B, or looked in 00:46:41.380 --> 00:46:42.640 terminal C? 00:46:42.640 --> 00:46:46.540 Figure out the Thevenin and Norton parameters, and see if 00:46:46.540 --> 00:46:47.790 there's an error in the table. 00:52:09.860 --> 00:52:11.213 So how many errors are in the table? 00:52:17.670 --> 00:52:20.890 About 50% correct again. 00:52:20.890 --> 00:52:22.640 So which entry don't you like. 00:52:25.604 --> 00:52:27.086 AUDIENCE: 2D? 00:52:27.086 --> 00:52:30.060 PROFESSOR: That's exactly right, so 2D is wrong. 00:52:30.060 --> 00:52:32.002 How do I figure out 1A? 00:52:32.002 --> 00:52:35.180 How do I figure out V0 for the a circuit? 00:52:39.784 --> 00:52:43.600 So the definition of V0 is the open circuit voltage. 00:52:43.600 --> 00:52:47.220 So I need to figure out, for the a circuit, how big would 00:52:47.220 --> 00:52:51.770 be the voltage across A if there was no current flowing 00:52:51.770 --> 00:52:55.430 in the leg of A. Everybody clear on that? 00:52:55.430 --> 00:52:57.510 So the thing that I would do is, I would think about, 00:52:57.510 --> 00:52:59.470 there's no connection here. 00:52:59.470 --> 00:53:00.440 What's the voltage here? 00:53:00.440 --> 00:53:02.810 So how would I calculate that? 00:53:02.810 --> 00:53:03.613 Yeah. 00:53:03.613 --> 00:53:06.028 AUDIENCE: You would use the current-divider relationship 00:53:06.028 --> 00:53:10.616 to figure out that current flowing through that is going 00:53:10.616 --> 00:53:11.824 to be 4 Amps. 00:53:11.824 --> 00:53:14.722 PROFESSOR: Precisely, so first I need to take-- here are the 00:53:14.722 --> 00:53:18.065 current sources, so I have two resistor lengths, so that's a 00:53:18.065 --> 00:53:21.240 perfect set up for the current divider. 00:53:21.240 --> 00:53:25.380 So the amount of current that goes in this leg compared to 00:53:25.380 --> 00:53:30.360 that leg is the ratio of this resistance to the sum of 00:53:30.360 --> 00:53:32.460 resistances, right? 00:53:32.460 --> 00:53:35.020 And if you work that out, you're going to get 4 Amps 00:53:35.020 --> 00:53:37.550 coming through here. 00:53:37.550 --> 00:53:40.870 So then after you know the current through this leg, it's 00:53:40.870 --> 00:53:43.700 an easy matter to take the current and turn it into a 00:53:43.700 --> 00:53:48.590 voltage, so the voltage at the a port is 4 Amps times 5 Ohms 00:53:48.590 --> 00:53:49.945 is 20 Volts. 00:53:49.945 --> 00:53:51.195 Is that clear? 00:53:53.810 --> 00:53:58.650 And similarly, but with a different answer, the voltage 00:53:58.650 --> 00:54:03.860 at the B terminal is the same 4 Amps, but now times 10 Ohms. 00:54:03.860 --> 00:54:06.040 So that's how we got 40. 00:54:06.040 --> 00:54:08.940 And at C, it's the same 4 Amps, but now it's times the 00:54:08.940 --> 00:54:12.150 sum, 4 times 15 is 60. 00:54:12.150 --> 00:54:13.630 Everybody's happy about that? 00:54:13.630 --> 00:54:18.330 So the point is that when you generate an equivalent 00:54:18.330 --> 00:54:22.610 circuit, it depends upon which set of terminals you're using. 00:54:22.610 --> 00:54:25.050 You can't just take a circuit and say, give me the Thevenin 00:54:25.050 --> 00:54:25.540 Equivalent. 00:54:25.540 --> 00:54:27.040 You have to say, give me the Thevenin 00:54:27.040 --> 00:54:29.190 equivalent looking somewhere. 00:54:29.190 --> 00:54:32.390 So I could look in the A port, the B port, or the C port, and 00:54:32.390 --> 00:54:34.850 I get different V0's. 00:54:34.850 --> 00:54:36.100 How would I compute the I0? 00:54:42.188 --> 00:54:45.182 Short circuit current-- 00:54:45.182 --> 00:54:49.370 so what I would do is I would short this out. 00:54:49.370 --> 00:54:52.543 When I do that, all the current flows in this leg and 00:54:52.543 --> 00:54:55.550 one of the current flows that leg. 00:54:55.550 --> 00:54:57.890 So that means I have, equivalently, 10 Ohms in 00:54:57.890 --> 00:55:00.440 parallel with 10 Ohms. 00:55:00.440 --> 00:55:03.310 Then by the current divider, how much current 00:55:03.310 --> 00:55:06.230 goes down one leg? 00:55:06.230 --> 00:55:07.480 Half of it. 00:55:09.910 --> 00:55:14.380 I get a different answer over here, because now I short out 00:55:14.380 --> 00:55:17.660 that node, which shorts out that resistor, so now I get a 00:55:17.660 --> 00:55:20.530 different ratio of resistors. 00:55:20.530 --> 00:55:22.880 It's not the same as the first, so I know that this 00:55:22.880 --> 00:55:25.640 answer can't be 5. 00:55:25.640 --> 00:55:28.630 And, in fact, if you work it out, the answer is 20 over 3. 00:55:31.860 --> 00:55:38.140 And finally, if you short here, then you know that that 00:55:38.140 --> 00:55:42.600 short circuit shorts out both this series 00:55:42.600 --> 00:55:44.940 combination and that one. 00:55:44.940 --> 00:55:47.270 So all of the 10 Amps goes through that, so 00:55:47.270 --> 00:55:48.790 you get that 10 Amps. 00:55:48.790 --> 00:55:52.150 Then how do you get R0, which is the ratio of V0 over I0. 00:55:55.370 --> 00:55:58.560 So if you take V0 over I0, you get 4. 00:55:58.560 --> 00:56:01.460 Here if you do right answer, you get 6, and here if you do 00:56:01.460 --> 00:56:03.840 that you get 6. 00:56:03.840 --> 00:56:06.380 The point is, that the Thevenin 00:56:06.380 --> 00:56:08.680 equivalent you get is different. 00:56:08.680 --> 00:56:11.320 The Norton equivalent that you get is different, depending on 00:56:11.320 --> 00:56:12.570 which ports you're looking at. 00:56:15.460 --> 00:56:20.980 OK there's two reasons for thinking about this. 00:56:20.980 --> 00:56:21.820 One-- 00:56:21.820 --> 00:56:24.530 so why am I thinking about all of these equivalent circuits? 00:56:24.530 --> 00:56:28.120 So I wanted to have an abstraction that was useful 00:56:28.120 --> 00:56:29.950 for thinking about how parts interact. 00:56:32.520 --> 00:56:35.800 I wanted a way of thinking about what would happen if I 00:56:35.800 --> 00:56:40.300 changed the load on the circuit without having to 00:56:40.300 --> 00:56:42.920 recalculate all the voltages and currents 00:56:42.920 --> 00:56:44.100 throughout the circuit. 00:56:44.100 --> 00:56:47.260 And so this Thevenin and Norton idea is a 00:56:47.260 --> 00:56:48.820 way of doing that. 00:56:48.820 --> 00:56:51.290 That's important from a practical sense, because when 00:56:51.290 --> 00:56:52.660 you buy a part-- 00:56:52.660 --> 00:56:55.910 when you buy an electronic part, they tell you how it 00:56:55.910 --> 00:56:58.940 works by telling you the Thevenin equivalent or the 00:56:58.940 --> 00:57:01.150 Norton equivalent, or whatever is the easy way 00:57:01.150 --> 00:57:03.280 to think about it. 00:57:03.280 --> 00:57:05.030 So there's a practical reason-- 00:57:05.030 --> 00:57:07.970 when you buy an op-amp, they tell you how good the op-amp 00:57:07.970 --> 00:57:10.960 is by telling you how big is the equivalent resistance at 00:57:10.960 --> 00:57:12.370 the output. 00:57:12.370 --> 00:57:15.470 So it has a practical value, because it lets you-- it's the 00:57:15.470 --> 00:57:19.020 way you specify an electronic part. 00:57:19.020 --> 00:57:21.910 You can't -- when the manufacturer makes a part, 00:57:21.910 --> 00:57:24.670 they can't know what you're going to do with it. 00:57:24.670 --> 00:57:28.180 So they tell you how it works by telling you something about 00:57:28.180 --> 00:57:30.030 the equivalent circuit. 00:57:30.030 --> 00:57:32.480 There's also a different reason for thinking about 00:57:32.480 --> 00:57:36.110 this, and that is because it's conceptually simplifying to 00:57:36.110 --> 00:57:39.390 think about Thevenin and Norton equivalents. 00:57:39.390 --> 00:57:41.530 So here's an example that's very much like the first 00:57:41.530 --> 00:57:43.340 problem that I worked out. 00:57:43.340 --> 00:57:45.670 What would be the effect of closing this switch 00:57:45.670 --> 00:57:48.770 on the current I? 00:57:48.770 --> 00:57:51.990 We solved the problem very much like this last time, and 00:57:51.990 --> 00:57:55.170 one way you can solve it is you figure out I in two cases, 00:57:55.170 --> 00:57:57.190 when the switch is open and when the switch is closed. 00:57:57.190 --> 00:57:59.010 And you figure out whether went up or down, and you know 00:57:59.010 --> 00:57:59.990 the answer. 00:57:59.990 --> 00:58:01.910 The point is, that if you think about this in terms of 00:58:01.910 --> 00:58:04.380 equivalent circuits, it's completely trivial. 00:58:07.110 --> 00:58:08.730 If I think about what would happen-- 00:58:08.730 --> 00:58:12.740 I'm interested in what happens at I, what I do is split the 00:58:12.740 --> 00:58:15.203 circuit into two pieces, the stuff to the left of I, and 00:58:15.203 --> 00:58:17.070 the stuff to the right to of I. I make a Thevenin 00:58:17.070 --> 00:58:19.500 equivalent for both of them, and then I jam together the 00:58:19.500 --> 00:58:22.780 two Thevenin equivalents. 00:58:22.780 --> 00:58:27.230 What I get, in detail, is showed here. 00:58:27.230 --> 00:58:31.240 So if I look left, I see a 20 Volt source with 00:58:31.240 --> 00:58:34.070 a 4, 4, and a 2. 00:58:34.070 --> 00:58:37.970 That has a Thevenin equivalent that is showed here, which is 00:58:37.970 --> 00:58:40.670 independent of the state of the switch. 00:58:40.670 --> 00:58:43.930 So the same Thevenin occurs on the two sides, switch open, or 00:58:43.930 --> 00:58:46.070 switch closed. 00:58:46.070 --> 00:58:49.040 If i make a Thevenin over here, there's no sources. 00:58:49.040 --> 00:58:51.670 So it's just going to be a resistance. 00:58:51.670 --> 00:58:54.133 So when the switch is open, I have to 2 Ohms, when the 00:58:54.133 --> 00:58:55.383 switch is closed, I have 1 Ohm. 00:58:58.910 --> 00:59:03.620 I don't even need to figure out what are these part values 00:59:03.620 --> 00:59:07.490 to see that, if I make this resistance smaller, which 00:59:07.490 --> 00:59:12.096 happens if I close the switch, the current goes up. 00:59:12.096 --> 00:59:15.370 I mean it's true that I showed here that the Thevenin voltage 00:59:15.370 --> 00:59:17.370 is 10 and the Thevenin resistor is 4. 00:59:17.370 --> 00:59:19.720 I don't even care. 00:59:19.720 --> 00:59:21.760 Regardless of what the Thevenin voltage was, 00:59:21.760 --> 00:59:23.800 regardless of what the Thevenin current is, it's 00:59:23.800 --> 00:59:26.270 going to be the same when the switch is open and closed. 00:59:26.270 --> 00:59:29.710 And that's a powerful statement. 00:59:29.710 --> 00:59:33.050 I know it's the same, so when I close this which, all I've 00:59:33.050 --> 00:59:35.990 really done is I've made that resistor smaller. 00:59:35.990 --> 00:59:40.620 The net series resistance is down, the current is up. 00:59:40.620 --> 00:59:43.610 So I have two reasons for thinking about Thevenin and 00:59:43.610 --> 00:59:44.290 Norton equivalence. 00:59:44.290 --> 00:59:48.720 One is practical value, the other is conceptual 00:59:48.720 --> 00:59:49.760 simplicity. 00:59:49.760 --> 00:59:52.440 It lets you simplify way you think about a circuit, and 00:59:52.440 --> 00:59:54.740 it's a way of gaining intuition without ever even 00:59:54.740 --> 00:59:56.310 solving the equations. 00:59:56.310 --> 01:00:00.120 Most circuit designers don't solve certain equations. 01:00:00.120 --> 01:00:03.020 They know what it's going to do just by looking at it, and 01:00:03.020 --> 01:00:05.150 this is the kind of reasoning that they use. 01:00:08.130 --> 01:00:10.310 OK, there's one more topic-- 01:00:10.310 --> 01:00:13.520 the idea of Thevenins and Nortons really derived from 01:00:13.520 --> 01:00:16.320 linear algebra. 01:00:16.320 --> 01:00:19.730 The basic parts of most interest are linear. 01:00:19.730 --> 01:00:21.710 And when you put together a system out of linear parts, 01:00:21.710 --> 01:00:24.410 you get a linear system. 01:00:24.410 --> 01:00:28.090 There's one more consequence of linearity that is terribly 01:00:28.090 --> 01:00:32.020 useful, and that's the idea of super position. 01:00:32.020 --> 01:00:35.000 If you have a system of equations that is linear, and 01:00:35.000 --> 01:00:39.710 if you have multiple sources, things are very simple. 01:00:39.710 --> 01:00:42.050 You can see that over here. 01:00:42.050 --> 01:00:45.310 If I had multiple sources, the system of equations that I get 01:00:45.310 --> 01:00:47.540 wouldn't look quite like this. 01:00:47.540 --> 01:00:49.720 These terms, the constant terms, they're the ones that 01:00:49.720 --> 01:00:51.810 come from the drives, the voltage 01:00:51.810 --> 01:00:53.120 sources, the current sources. 01:00:53.120 --> 01:00:56.040 So if I had two drives, I would get something else here. 01:00:56.040 --> 01:00:59.790 I would get plus a at N+1 for example-- 01:00:59.790 --> 01:01:02.650 source one, source two. 01:01:02.650 --> 01:01:06.471 And I would get source one, source two. 01:01:06.471 --> 01:01:10.850 So the idea is that, if I have a system that has multiple 01:01:10.850 --> 01:01:16.200 sources, I know from the structure of the linear 01:01:16.200 --> 01:01:20.846 equations, there is just more constant terms. 01:01:20.846 --> 01:01:22.480 OK, well what's that mean? 01:01:22.480 --> 01:01:25.720 That means you can just use linear algebra to see that the 01:01:25.720 --> 01:01:29.790 answer to this problem is the sum of the answers to that 01:01:29.790 --> 01:01:33.970 problem plus the answers to that problem. 01:01:33.970 --> 01:01:37.050 That's just linear algebra. 01:01:37.050 --> 01:01:40.130 What that means in terms of circuits is, I can figure out 01:01:40.130 --> 01:01:42.640 the response to a circuit by turning on the 01:01:42.640 --> 01:01:44.590 sources one at a time. 01:01:44.590 --> 01:01:47.360 That's called superposition. 01:01:47.360 --> 01:01:50.440 And generally speaking, it's a lot easier than solving the 01:01:50.440 --> 01:01:52.630 circuit out the long way. 01:01:52.630 --> 01:01:57.180 So here I've got two sources So say I wanted to compute I, 01:01:57.180 --> 01:02:01.040 in response to V0 and I0. 01:02:01.040 --> 01:02:05.930 What I would do, is I would turn off the I0 source, and 01:02:05.930 --> 01:02:08.420 calculate the voltage that results just 01:02:08.420 --> 01:02:10.400 from the voltage source. 01:02:10.400 --> 01:02:13.610 That's the same as setting this to 0 and finding the 01:02:13.610 --> 01:02:14.860 response to this. 01:02:18.240 --> 01:02:22.820 So if I want to set I to 0, the way you set I to be 0 is 01:02:22.820 --> 01:02:25.790 open circuit. 01:02:25.790 --> 01:02:29.160 If you open circuit something, there's no current going to 01:02:29.160 --> 01:02:30.410 flow through it. 01:02:30.410 --> 01:02:34.960 So I replace the current source with an open circuit, 01:02:34.960 --> 01:02:38.860 and compute the i that would be result when the current 01:02:38.860 --> 01:02:40.970 source isn't there. 01:02:40.970 --> 01:02:42.540 OK, well that's easy. 01:02:42.540 --> 01:02:44.480 If their current source we're not there, this 01:02:44.480 --> 01:02:45.500 would be open circuit. 01:02:45.500 --> 01:02:48.630 The current, i, would just be the total voltage going 01:02:48.630 --> 01:02:50.770 through R1 and R2. 01:02:50.770 --> 01:02:56.890 So the answer, I1, the first component of the current I, 01:02:56.890 --> 01:03:00.840 would be V0, the result of the voltage source divided by the 01:03:00.840 --> 01:03:02.850 sum of R1 and R2. 01:03:02.850 --> 01:03:03.840 Now that's not the whole answer. 01:03:03.840 --> 01:03:07.580 That would be the whole answer if I0 weren't there. 01:03:07.580 --> 01:03:11.280 So now I have to worry about the other case. 01:03:11.280 --> 01:03:13.810 What if only I0 were there? 01:03:13.810 --> 01:03:16.860 Well now I have to set V to be 0. 01:03:16.860 --> 01:03:20.060 Well setting V to 0 is not open circuiting it. 01:03:20.060 --> 01:03:22.806 You can't just reach in, grab the voltage source, and throw 01:03:22.806 --> 01:03:27.220 it away, because that'll make the current 0. 01:03:27.220 --> 01:03:31.900 If I want the voltage to be 0, I have to short circuit it. 01:03:31.900 --> 01:03:35.610 So what I do then, is I leave the current source alone, and 01:03:35.610 --> 01:03:38.030 I replace the voltage source with a short circuit, 01:03:38.030 --> 01:03:41.780 guaranteeing that V is 0. 01:03:41.780 --> 01:03:46.160 Then I ask, how big is I2, the component of I that results 01:03:46.160 --> 01:03:48.730 from the current source? 01:03:48.730 --> 01:03:52.680 Well if I've short circuited this, then I just get a 01:03:52.680 --> 01:03:55.200 current divider. 01:03:55.200 --> 01:03:58.720 This current has to do with how readily the current 01:03:58.720 --> 01:04:01.030 divides between R1 and R2. 01:04:01.030 --> 01:04:04.170 The amount that goes through the R1 side is in proportion 01:04:04.170 --> 01:04:07.670 to R2 make R2 bigger, more of it goes through R1. 01:04:07.670 --> 01:04:10.120 Standard current divider, except for the 01:04:10.120 --> 01:04:11.370 slippery minus sign. 01:04:14.460 --> 01:04:17.960 So if I only had the current source, the current I2 would 01:04:17.960 --> 01:04:22.150 have been current divider operating I0 with the slippery 01:04:22.150 --> 01:04:23.280 minus sign. 01:04:23.280 --> 01:04:26.770 So that means, by superposition, that the result 01:04:26.770 --> 01:04:29.475 of having both sources on is just the sum of those answers. 01:04:32.390 --> 01:04:35.570 That's a very big simplification. 01:04:35.570 --> 01:04:39.050 If you've got multiple sources in the circuit, you can think 01:04:39.050 --> 01:04:41.540 about them all at once. 01:04:41.540 --> 01:04:44.410 And if you really good at writing linear equations and 01:04:44.410 --> 01:04:48.760 solving them with pencils, you'll get the right answer. 01:04:48.760 --> 01:04:51.820 But there's an enormous simplification if you just 01:04:51.820 --> 01:04:55.950 turn off all but one, and do them one at a time. 01:04:55.950 --> 01:04:57.680 So here's a problem. 01:04:57.680 --> 01:05:00.770 Here's a very simple circuit, compute V 01:05:00.770 --> 01:05:02.940 by using super position. 01:05:02.940 --> 01:05:04.350 V is the voltage across the resistor. 01:05:04.350 --> 01:05:07.420 How big would V be if you used the idea of super position? 01:06:52.790 --> 01:06:54.040 So what's the answer? 01:07:03.190 --> 01:07:08.010 50% correct, roughly speaking. 01:07:08.010 --> 01:07:09.200 OK I want to use superposition. 01:07:09.200 --> 01:07:12.580 So how big would the voltage v be, if all I had was the 01:07:12.580 --> 01:07:14.910 voltage source? 01:07:14.910 --> 01:07:15.430 One. 01:07:15.430 --> 01:07:17.000 How big would the voltage be if i only 01:07:17.000 --> 01:07:19.106 had the current source? 01:07:19.106 --> 01:07:21.510 Ah, got half of you. 01:07:21.510 --> 01:07:24.250 That explains the 50% correct. 01:07:24.250 --> 01:07:25.693 How big would the voltage be, if I only 01:07:25.693 --> 01:07:27.970 had the current source? 01:07:27.970 --> 01:07:29.560 It's tempting to say (1). 01:07:29.560 --> 01:07:34.390 I say tempting, because that's the wrong answer. 01:07:34.390 --> 01:07:36.810 [LAUGHTER] 01:07:36.810 --> 01:07:39.530 What is wrong about the answer (1)? 01:07:39.530 --> 01:07:44.464 What does the voltage do to the current source? 01:07:44.464 --> 01:07:46.776 The voltage due to the current source is the voltage that the 01:07:46.776 --> 01:07:48.020 current source would have generated if the voltage 01:07:48.020 --> 01:07:50.730 source weren't there. 01:07:50.730 --> 01:07:52.335 If the voltage source weren't there, then-- 01:07:55.500 --> 01:07:57.050 half of you got it right, so half of you 01:07:57.050 --> 01:07:58.300 can shout the answer. 01:08:01.340 --> 01:08:04.600 So if the voltage source weren't there, the voltage 01:08:04.600 --> 01:08:05.920 source would be 0. 01:08:05.920 --> 01:08:09.920 If the voltage source were 0, then V would be 0. 01:08:09.920 --> 01:08:13.290 So if the voltage due to the voltage source is 1, the 01:08:13.290 --> 01:08:15.910 voltage due to the current source is 0. 01:08:15.910 --> 01:08:17.310 The sum of the two is 1. 01:08:17.310 --> 01:08:18.180 The answer is 1. 01:08:18.180 --> 01:08:19.899 OK? 01:08:19.899 --> 01:08:21.149 Make sense? 01:08:24.420 --> 01:08:26.689 So, very closely related problem. 01:08:26.689 --> 01:08:28.510 What's the current I? 01:08:28.510 --> 01:08:29.760 Solve that by superposition. 01:10:08.110 --> 01:10:12.800 So how big is the current I according to superposition? 01:10:12.800 --> 01:10:18.244 Wonderful, so how big is the current i generated by the 01:10:18.244 --> 01:10:21.070 voltage source? 01:10:21.070 --> 01:10:22.892 1. 01:10:22.892 --> 01:10:26.286 How big is the I generated by the current source? 01:10:26.286 --> 01:10:28.980 Negative 1, the sum is 0. 01:10:28.980 --> 01:10:30.990 All right, so those two problems were trivial by 01:10:30.990 --> 01:10:31.395 superposition. 01:10:31.395 --> 01:10:33.840 They're not too hard by non-superposition. 01:10:33.840 --> 01:10:36.280 But the point is, that they're trivial by super position. 01:10:36.280 --> 01:10:38.220 So what we saw today-- 01:10:38.220 --> 01:10:41.700 the goal for today was to generate some abstractions 01:10:41.700 --> 01:10:44.730 that let you think about the way parts interact with each 01:10:44.730 --> 01:10:48.480 other, because that's a central issue when you're 01:10:48.480 --> 01:10:51.700 thinking about circuit design And we built the idea of 01:10:51.700 --> 01:10:54.100 Thevenins and Nortons and super positions. 01:10:54.100 --> 01:10:55.740 That was the basic idea. 01:10:55.740 --> 01:10:57.450 And the sub-theme-- 01:10:57.450 --> 01:11:01.450 or maybe I should say the major theme is really that. 01:11:01.450 --> 01:11:03.110 It's the importance of linear algebra. 01:11:03.110 --> 01:11:07.610 So the take home message is probably, take 18.06. 01:11:07.610 --> 01:11:11.250 So this was all about the application of 18.06 to the 01:11:11.250 --> 01:11:13.230 solving of circuits. 01:11:13.230 --> 01:11:14.480 See you later.